3. Chocolates without simple formulas for L-state

We define a new type of chocolate precisely. Please compare this to the chocolate in Fig 3.1, since an example of this definition is the chocolate of Fig 3.1.

We connect points and with segments for , and we also connect points and with segments for . We make a trapezoid with four points , , , , and color this trapezoid with red. We color all the other parts of the triangle with green. All the segments are in black.

green polygons are sweet chocolate that can be eaten, and the red trapezoid is the bitter chocolate that cannot be eaten.

You cut parallel to the segment on the left side of the red trapezoid.

You cut horizontally above the red (bitter) trapezoid.

You cut parallel to the segment on the right side of the red trapezoid.

Therefore it is proper to represent these chocolates with , where x,y,z stand for the maximum numbers of times that we can cut these chocolate in each direction. For example in Fig 3.1 we can cut 2 times at most parallel to the segment on the left side of the red trapezoid, 5 times at most horizontally and 5 times at most parallel to the segment on the right side of the red trapezoid. Therefore , and . Therefore we represent the chocolate in Fig 3.1 with the coordinates .

Although we do not have a simple formula for the set of L states in the
case of this chocolate game, we have some prediction for the set of L
states.

First we define six functions
and
for natural numbers
.

By using Definition 3.2 we present a prediction for the set of L states.

There is another prediction for the set of L states.

Instead of proving these prediction, the authors present the calculation by computer algebra system Mathematica.

By calculation of Mathematica these predictions seem to be true.

ppo[ss_] := Block[{k}, k = 1;

al = Flatten[Table[{a, b, c}, {a, 0, ss}, {b, 0, ss}, {c, 0, ss}],

2];

allcases = Select[al, (1/k) (#[[3]]) >= #[[2]] &];

move[z_] := Block[{p}, p = z;

Union[Table[{t1, p[[2]], p[[3]]}, {t1, 0, p[[1]] - 1}],

Table[{p[[1]], t2, p[[3]]}, {t2, 0, p[[2]] - 1}],

Table[{p[[1]], Min[Floor[t3/k], p[[2]]], t3}, {t3, 0, p[[3]] - 1}]

]

];

Mex[L_] := Min[Complement[Range[0, Length[L]], L]];

Gr[pos_] := Gr[pos] = Mex[Map[Gr, move[pos]]];

pposition[0] = Select[allcases, Gr[#] == 0 &]]

AA[n_] :=

Union[Join[

Table[{6 n - 1, 4 n - 4 k - 1, 4 n + 2 k - 1}, {k, 0, n - 1}],

Table[{6 n - 1, 4 n - 4 k - 2, 4 n + 2 k}, {k, 0, n - 1}],

Table[{6 n - 1, 4 k, 6 n + 2 k - 1}, {k, 0, n - 1}],

Table[{6 n - 1, 4 k + 1, 6 n + 2 k}, {k, 0, n - 1}]]];

BB[n_] :=

Union[Join[Table[{6 n, 4 n - 4 k, 4 n + 2 k}, {k, 0, n}],

Table[{6 n, 4 n - 4 k - 3, 4 n + 2 k + 1}, {k, 0, n - 1}],

Table[{6 n, 4 k + 2, 6 n + 2 k + 1}, {k, 0, n - 1}],

Table[{6 n, 4 k + 3, 6 n + 2 k + 2}, {k, 0, n - 1}]]];

CC[n_,m_] :=

Union[Join[

Table[{6 n + 1, 4 n - 4 k - 1, 4 n + 2 k + 1}, {k, 0, n - 1}],

Table[{6 n + 1, 4 n - 4 k - 2, 4 n + 2 k + 2}, {k, 0, n - 1}],

Table[{6 n + 1, 4 k, 6 n + 2 k + 1}, {k, 0, n}],

Table[{6 n + 1, 4 k + 1, 6 n + 2 k + 2}, {k, 0, n - 1}],

Table[{6 n + 1, k + 4 n + 1, k + 8 n + 2}, {k, 0, n + m}]]];

DD[n_] :=

Union[Join[

Table[{6 n + 2, 4 n - 4 k + 1, 4 n + 2 k + 1}, {k, 0, n}],

Table[{6 n + 2, 4 n - 4 k, 4 n + 2 k + 2}, {k, 0, n}],

Table[{6 n + 2, 4 k + 2, 6 n + 2 k + 3}, {k, 0, n - 1}],

Table[{6 n + 2, 4 k + 3, 6 n + 2 k + 4}, {k, 0, n - 1}]]];

EE[n_] :=

Union[Join[

Table[{6 n + 3, 4 n + 2 - 4 k, 4 n + 2 k + 2}, {k, 0, n}],

Table[{6 n + 3, 4 n - 4 k - 1, 4 n + 2 k + 3}, {k, 0, n - 1}],

Table[{6 n + 3, 4 k, 6 n + 2 k + 3}, {k, 0, n}],

Table[{6 n + 3, 4 k + 1, 6 n + 2 k + 4}, {k, 0, n}]]];

FF[n_,m_] :=

Union[Join[

Table[{6 n + 4, 4 n - 4 k + 1, 4 n + 2 k + 3}, {k, 0, n}],

Table[{6 n + 4, 4 n - 4 k, 4 n + 2 k + 4}, {k, 0, n}],

Table[{6 n + 4, 4 k + 2, 6 n + 2 k + 5}, {k, 0, n}],

Table[{6 n + 4, 4 k + 3, 6 n + 2 k + 6}, {k, 0, n - 1}],

Table[{6 n + 4, k + 4 n + 3, k + 8 n + 6}, {k, 0, n + m}]]];

na[tt_] :=

Union[Flatten[

Table[Join[CC[n, m], FF[n, m]], {n, 0, tt}, {m, 0, tt}], 2],

Flatten[Table[Join[AA[n], BB[n], DD[n], EE[n]], {n, 0, tt}], 1]];

dat[nn_] :=

Join[Flatten[

Table[{Ceiling[3 a/2] + 2 n, a, a + 2 n}, {a, 0, nn}, {n, 0, nn}],

1], Flatten[

Table[If[

n <= Floor[a/2], {3 n + 1, a, a + 1 + 2 n},

{2 n + Floor[a/2] + 1, a, a + 1 + 2 n}], {n, 0, nn}, {a, 0,

nn}], 1]];

In[1]:= Complement[ppo[8], dat[9]]

Out[1]= {}

In[2]:= Complement[dat[9], na[8]]

Out[2]= {}

In[3]:= Complement[na[8], ppo[90]]

Out[3]= {}

In[4]:= Complement[ppo[90], dat[90]]

Out[4]= {}